If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
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If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
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One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is
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Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be $$\widehat{i}+2\widehat{j}+\widehat{k},\widehat{i}+3\widehat{j}-2\widehat{k}$$ and $$2\widehat{i}+\widehat{j}-\widehat{k}$$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through of the triangle ABC at the point . If the length of AD is $$\frac{\sqrt{110}}{3}$$ and the volume of the tetrahedron is $$\frac{\sqrt{805}}{6\sqrt{2}}$$, then the position vector of is
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If A, B and $$(adj (A^{-1})+adj(B^{-1}))$$ are non-singular matrices of same order, then the inverse of $$A(adj(A^{-1}+adj(B^{-1}))^{-1}B$$, is equal to
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Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is
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Let a curve y=f(x) pass through the points (0,5) and $$(\log_{e}2,k)$$ . If the curve satisfies the differential equation $$2(3+y)e^{2x}dx-(7+e^{2x})dy=0$$ , then k is equal to
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If the function $$f(x)=\begin{cases}\frac{2}{x}\{\sin((k_1+1)x)+\sin(k_2-1)x\}, & x<0 \\4, & x=0 \\\frac{2}{x}\log_e\left(\frac{2+k_1x}{2+k_2x}\right), & x>0\end{cases}$$ is continuous at x=0, then $$k_1^2+k_2^2$$ is equal to
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If the line 3x-2y+12=0 intersects the parabola $$4y=3x^{2}$$ At the points A and B , then at the vertex of the parabola, the line segment AB subtends an angle equal to
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Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line $$\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}$$. Then the area of the right angled triangle PQR , where R is the point (3,-2,1),is
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Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that $$\frac{\text{lenght of arc AB}}{\text{lenght of arc BC}}=\frac{1}{5}$$,and $$\overrightarrow{OC}=\alpha\overrightarrow{OA}+\beta\overrightarrow{OB}$$, then $$\alpha +\sqrt{2}(\sqrt{3}-1)\beta$$ is equal to
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Let $$f(x)=\log_{e}x$$ and $$g(x)=\frac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$$. Then the domain of $$f \circ g$$ is
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If the system of equations $$(\lambda-1)x+(\lambda-4)y+\lambda z=5 \\\lambda x+(\lambda-1)y+(\lambda-4)z=7 \\ (\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$$ has infinitely many solutions, then $$\lambda^{2}+\lambda$$ is equal to
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The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
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Let $$R=\left\{(1,2),(2,3),(3,3)\right\}$$ be a relation defined on the set $$\left\{1,2,3,4\right\}$$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:
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Let the area of a $$\triangle PQR$$ with vertices P(5,4), Q(-2,4) and R(a,b) be 35 square units. If its orthocenter and centroid are $$O(2,\frac{14}{5})$$ and C(c,d) respectively, then c+2d is equal to
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The value of $$\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{((log_{e}x)^{2}+1)^{-1}}}{e^{((log_{e}x)^{2}+1)^{-1}}+e^{((6-\log_{e}x)^{2}+1)^{-1}}}\right)dx$$ is
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Let $$\mid\frac{\overline{z}-i}{2\overline{z}+i}\mid=\frac{1}{3}, z\in C$$, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and $$(\alpha,0)$$ is 11 square units, then $$\alpha^{2}$$ equals:
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The value of $$\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ}\cot 70^{\circ}-1\right)$$ is
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Let $$I(x)=\int_{}^{} \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$$. If $$I(37)-I(24)=\frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}}-\frac{1}{c^{\frac{1}{13}}}\right),b,c\in N$$, then 3(b+c) is equal to
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If $$\frac{\pi}{2}\leq x\leq \frac{3\pi}{4}$$, then $$\cos^{-1}\left(\frac{12}{13}\cos x+\frac{5}{13}\sin x\right)$$ is equal to
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Let the circle touch the line x-y+1=0, have the centre on the positive x -axis, and cut off a chord of length $$\frac{4}{\sqrt{13}}$$ along the line -3x+2y=1. Let H be the hyperbola $$\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$$ , whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then $$2\alpha^{2}+3\beta^{2}$$ is equal to ______
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If the equation $$a(b-c)x^{2}+b(c-a)x+c(a-b)=0$$ has equal roots, where a+c=15 and $$b=\frac{36}{5}$$, then $$a^{2}+c^{2}$$ is equal to
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If the set of all values of a, for which the equation $$5x^{3}-15x-a=0$$ has three distinct real roots, is the interval $$(\alpha, \beta)$$, then $$\beta-2\alpha $$ is equal to ______
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The sum of all rational terms in the expansion of $$(1+2^{1/2}+3^{1/2})^{6}$$ is equal to
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If the area of the larger portion bounded between the curves $$x^{2}+y^{2}=25$$ and y=|x-1| is $$\frac{1}{4}(b\pi+c),b,c\in N$$, then b+c is equal to
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